70-300 or converter?

I currently have the 70-200 f4L (non-is) and would like to get some more reach. Would I be better off buying the Canon 70-300 or getting a 1.4 or even 2x converter for the 70-200?

My body is a 5d classic. I would like to get reach for airplane shots, sport events (daytime), etc.

My budget is in the $300-400 range.

Your 70/200 will AF Ok with a 1.4 TC and will give you 280mm f5.6 with only minimal IQ loss ( well my 70/200 f4 IS is OK) but it is not supposed to work with a 2*tc (its f8 and should not AF).
But-you will notice that I said it should not work with a 2*TC-there is a reason for this,I have a 40D and a 5Dc together with a 100/400.
If a use a Kenko 300 Pro DG 1.4TC with the pins taped the 100/400 will not AF with the 40D but it will quite happily AF with the 5D (f8) in good light,even though its not supposed to.
So if you can get hold of a 2*tc try it and see, it might work,but it will probably be slow to focus and the IQ may not be very good,the 1.4 will work fine.
Alternatively save up a bit more and try to pick up a used 100/400.